Page 72 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 7: Transformations and Conformal Mappings

7.1.2 Definitions Notes

(i) The points which coincide with their transforms under bilinear transformation are called
az b

its fixed points. For the bilinear transformation w = cz d , fixed points are given by

az b

w = z i.e. z = (1)
cz d

Since (1) is a quadratic in z and has in general two different roots, therefore, there are
generally two invariant points for a bilinear transformation.
(ii) If z , z , z , z are any distinct points in the z-plane, then the ratio
4
1
2
3
(z  z )(z  z )
3
2
4
1
(z , z , z , z ) = (z  z )(z  z )
1
2
4
3
2
1
4
3
is called cross ratio of the four points z , z , z , z . This ratio is invariant under a bilinear
1
2
4
3
transformation i.e.
(w , w , w , w ) = (z , z , z , z )
3
4
3
2
1
4
1
2
7.1.3 Transformation of a Circle
First we show that if p and q are two given points and K is a constant, then the equation
z p

z q = K, (1)

represents a circle. For this, we have
|z  p| = K |z  q| 2
2
2
2
 (z  p) z p  = K (z  q) z q 
2
 (z  p) ( z  p ) = K (z  q) (z q)
2


 zz  pz  p z pp = K (z z qz qz qq )

2
2
2
2
 (1  K ) z z (p  qK ) z (p q  K )z = K q q pp
2
2
2
2


 z z     p qK  z     p qK  z  |p|   K |q| 2 = 0 (2)
2 
2 
1 K
2
1 K 
1 K 
Equation (2) is of the form
zz bz bz c = 0 (c is being a real constant)



which always represents a circle.
Thus equation (2) represents a circle if K  1.
If K = 1, then it represents a straight line
| z  p | = | z  q |
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